OFFSET
1,2
COMMENTS
From Franklin T. Adams-Watters, Sep 13 2011: (Start)
If we use nonnegative integers instead of positive integers, we get this sequence shifted left (i.e., with offset 0).
The largest number that can be included in set of positive integers with mean n is the triangular number n*(n+1)/2 = A000217(n).
All values are odd. Sets including n are paired with the same set with n removed, with exception of {n}, as the empty set has no average.
(End)
LINKS
Martin Fuller, Table of n, a(n) for n = 1..200
FORMULA
Sum of coefficient of t^k x^(n*k) in Product_{i=1..n*k} (1+t*x^i) for k <= 2*n-1. - N. J. A. Sloane
From Martin Fuller, Sep 14 2023: (Start)
Constant term in formal Laurent series (Product_{i=1-n..n*(n-1)/2} (1+x^i)) - 1.
EXAMPLE
a(2) = 3 as there are three sets viz. {2}, {1,3}, {1,2,3}, each of which has the arithmetic mean 2.
a(3) = 9: the nine sets are {3}, {1, 5}, {2, 4}, {1, 2, 6}, {1, 3, 5}, {2, 3, 4}, {1, 2, 3, 6}, {1, 2, 4, 5}, {1, 2, 3, 4, 5}.
MAPLE
g := k->expand(mul(1+t*x^i, i=1..k)); A066571 := proc(n) local k; add(coeff(coeff(g(n*k), t, k), x, n*k), k=1..2*n-1); end;
MATHEMATICA
g[k_] := Expand[Product[1 + t*x^i, {i, 1, k}]]; a[n_] := Sum[Coefficient[ Coefficient[g[n*k], t, k], x, n*k], {k, 1, 2*n - 1}]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Feb 10 2018, translated from Maple *)
PROG
(Haskell)
a066571 n = f [1..] 1 n 0 where
f (k:ks) l nl x
| y > nl = 0
| y < nl = f ks (l + 1) (nl + n) y + f ks l nl x
| otherwise = if y `mod` l == 0 then 1 else 0
where y = x + k
-- Reinhard Zumkeller, Feb 13 2013
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Amarnath Murthy, Dec 19 2001
EXTENSIONS
Corrected and extended by N. J. A. Sloane, Dec 19 2001
More terms from Naohiro Nomoto, Jun 19 2002
More terms from David Wasserman, Sep 10 2002
More terms from Martin Fuller, Sep 14 2023
STATUS
approved